Operational and convolution properties of two-dimensional Fourier transforms in polar coordinates

For functions that are best described in terms of polar coordinates, the two-dimensional Fourier transform can be written in terms of polar coordinates as a combination of Hankel transforms and Fourier series-even if the function does not possess circular symmetry. However, to be as useful as its Cartesian counterpart, a polar version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives the requisite polar version of the standard Fourier operations. In particular, convolution-two dimensional, circular, and radial one dimensional-is discussed in detail. It is shown that standard multiplication/convolution rules do apply as long as the correct definition of convolution is applied.